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(B) Dimensional analysis is the most efficient way to solve this problem. The radius $R$ depends on energy $E$,density $\rho$,and time $t$.Let $R = k

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$t^{2/5}$

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(B) Dimensional analysis is the most efficient way to solve this problem. The radius $R$ depends on energy $E$,density $\rho$,and time $t$.Let $R = k E^a \rho^b t^c$,where $k$ is a dimensionless constant.The dimensions are: $[R] = L$,$[E] = ML^2T^{-2}$,$[\rho] = ML^{-3}$,$[t] = T$.Substituting these into the equation:$L = (ML^2T^{-2})^a (ML^{-3})^b (T)^c$$L^1 = M^{a+b} L^{2a-3b} T^{-2a+c}$Equating the powers of $M, L,$ and $T$:$a + b = 0 \Rightarrow b = -a$$2a - 3b = 1 \Rightarrow 2a - 3(-a) = 1 \Rightarrow 5a = 1 \Rightarrow a = 1/5$Thus,$b = -1/5$.$-2a + c = 0 \Rightarrow c = 2a = 2(1/5) = 2/5$.Therefore,$R \propto E^{1/5} \rho^{-1/5} t^{2/5}$,which implies $R \propto t^{2/5}$.

$A$ bomb explodes at time $t=0$ in a uniform,isotropic medium of density $\rho$ and releases energy $E$,generating a spherical blast wave. The radius $R$ of this blast wave varies with time $t$ as

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